Archimedes (287-212 BC) was a great ancient Greek philosopher, encyclopedia scientist, mathematician, physicist, and force scientist. He was the founder of static mechanics and fluid statics, and was known as the "father of mechanics". Archimedes, along with Gauss and Newton, are among the world's three greatest mathematicians. [1] Archimedes once said, "Give me a fulcrum, and I will be able to pry up the entire earth."
Archimedes established the fundamental principles of statics and fluid statics. Provide many methods for finding the center of gravity of geometric shapes, including the center of gravity of a shape enclosed by a parabola and its network parallel chords. Archimedes proved that the buoyancy of an object in a liquid is equal to the weight of the liquid it displaces, and this result was later known as the Archimedean principle. He also provided a criterion for the equilibrium stability of a parabolic rotating body floating in liquid. The machinery invented by Archimedes includes a water spiral for water diversion, a lever pulley mechanism that can pull a fully loaded ship, and an Earth Moon Sun motion model that can explain solar eclipses and lunar eclipse phenomena. But he believed that mechanical inventions were inferior to pure mathematics, so he did not write any works in this area. Archimedes also used the method of continuous segmentation to calculate the volume of ellipsoids, rotating projectiles, etc., which has already taken the form of integral calculation.

Birth of Characters
In 287 BC, Archimedes was born in a small village near Syracuse, Sicily. He was born into a noble family and was related to King Hieron of Syracuse, with a very wealthy family. Archimedes' father was an astronomer and mathematician, knowledgeable and humble. "Archimedes" in Greek means the great philosopher. Influenced by his family, Archimedes developed a strong interest in mathematics, astronomy, especially ancient Greek geometry from a young age.
When Archimedes was born, the glorious culture of ancient Greece had gradually declined, and the economic and cultural center had gradually shifted to the city of Alexandria in Egypt; On the other hand, the emerging Roman Republic on the Italian peninsula is also constantly expanding its influence; A new country, Carthage, also emerged in North Africa. Archimedes grew up in this era of alternating old and new forces, and the ancient city of Syracuse became a battleground for many forces. [1]
Academic Experience
In 267 BC, Archimedes was sent by his father to Alexandria, Egypt to study with the students of Euclid, Elatose and Canon. Alexandria, located at the mouth of the Nile River, was the center of knowledge, culture, and trade in the world at that time. It was a gathering place for scholars and talents, and was known as the "Capital of Wisdom" by the world. Research in literature, mathematics, astronomy, and medicine is highly developed.
Archimedes studied with many famous mathematicians in Alexandria, including the famous geometric master Euclid. Archimedes studied and lived here for many years, and he integrated the excellent cultural heritage of the East and ancient Greece, making a significant impact on his subsequent scientific career, laying the foundation for Archimedes to engage in scientific research in the future.

Defending the Motherland
The Second Punic War broke out between the Roman Republic and the Carthaginian Empire in North Africa in 218 BC. Syracuse, who was located on the island of Sicily, had always sought refuge with Rome. However, in 216 BC, Carthage suffered a great defeat against the Roman army. The new king of Syracuse (succeeded by the grandson of Hyveron II) immediately took the lead and allied with Carthage. The Roman Empire then sent General Marcelas to lead the army in attacking Syracuse both by sea and land.
The war broke out between Syracuse and the Roman Republic when Archimedes was old. Marcelas, the supreme commander of the Roman army, led the Roman army to surround the city where he lived and also occupied the seaport. Although Archimedes did not support war, he had to fulfill his responsibilities and defend his homeland. Archimedes saw that his homeland was in danger, and his sense of responsibility to protect his country prompted him to rise up against the enemy. Therefore, Archimedes racked his brains day and night to invent weapons to defend against the enemy.
● Throwers and cranes
Archimedes used the principle of leverage to create a stone thrower called a stone crossbow, which could throw large stones at Roman warships or use a transmitter to shoot spears and stones at Roman soldiers. Any enemy near the city wall could not escape his flying stones or javelins. Archimedes also invented various weapons to block the progress of the Roman army. According to some later records, he built a huge crane that could lift enemy warships into mid air and then drop them heavily to shatter them on the water surface.

Mirror spotlight
One day, the city of Syracuse was ambushed by the Roman army, and the young men and soldiers of Syracuse went to the front line. The city was left with only the elderly, women, and children, in a critical moment. At this moment, Archimedes stood up for his homeland.
Archimedes had women and children each take out their own mirrors from their homes and come to the coast together. The mirrors reflected strong sunlight onto the main sail of the enemy ship, and the reflections of thousands of mirrors gathered at a point on the sail. The sail burned, and the fire took advantage of the wind and grew stronger. The Romans were unaware of the details, thinking that Archimedes had invented a new weapon. He ran away in a panic.
These weapons caused panic and fear in the Roman army, even General Marcelas admitted with a bitter smile, "This is a war between the Roman fleet and Archimedes alone," and "Archimedes is a mythical giant with a hundred hands.".

The Death of a Great Man
In 212 BC, the ancient Roman army invaded Syracuse, and Archimedes was killed by Roman soldiers at the age of 75. Archimedes' body was buried on the island of Sicily, with a tombstone engraved with the shape of a cylindrical inscribed ball to commemorate his outstanding contributions in geometry.
Version 1: Roman soldiers break into Archimedes' residence and see an old man burying his head on the ground making geometric shapes. Archimedes tells the soldiers to wait and kill me. I cannot leave incomplete formulas for the world! Before he could finish speaking, the soldiers killed him. He died with regret.
Version 2: A Roman soldier suddenly appeared in front of him and ordered him to go to Marcelas, but was sternly rejected by Archimedes. As a result, Archimedes unfortunately died under the soldier's sword.
Version 3: Archimedes sits next to the broken stone wall, drawing a geometric shape on the sand. A Roman soldier ordered Archimedes to leave, and he made an arrogant gesture, saying, "Don't break my circle!" The Roman soldier was furious and immediately stabbed with a knife, killing the ancient scientist Archimedes.
Version 4: Roman soldiers break into Archimedes' residence and see an old man drawing and studying geometric problems on the ground in front of his house. Archimedes says, "Go away, don't touch my picture!" The soldier is very angry when he hears this, so he draws his knife and stabs Archimedes.
No matter how Archimedes died, the most regrettable thing was Marcelaus, the commander of the Roman army. Marcelaus executed the soldiers who had killed Archimedes as murderers. He held a grand funeral for Archimedes and built a tomb for him. According to Archimedes' wishes during his lifetime, the geometric figure of "cut inside a cylinder" was engraved on the tombstone.

Buoyancy principle
Brief description of buoyancy principle: The buoyancy obtained by an object in a liquid is equal to the weight of the liquid it discharges, that is, F=G (where F is the buoyancy exerted on the object and G is the gravity exerted on the object when it discharges the liquid). The deformation of this equation can be obtained
(In the equation) ρ For the density of the discharged liquid, g is the local gravitational acceleration, and V is the volume of the discharged liquid
Legend has it that King Syracuse had his craftsmen make a pure gold crown for him. But after it was done, the king suspected that the gold crown made by the craftsman was not pure gold. The craftsman embezzled the gold, but could not destroy the crown, and this gold crown was indeed as heavy as the pure gold originally given to the goldsmith. This question stumped the king and the ministers. On the suggestion of a minister, the king invited Archimedes to examine the crown.
At first, Archimedes was at a loss for this issue. One day, he was taking a shower at home. When he sat in the bathtub and saw water overflowing, he suddenly thought of a method to determine the volume of the golden crown by measuring the displacement of solids in water. He excitedly jumped out of the bathtub, didn't even bother putting on his clothes, and ran out, shouting loudly, "Yurika! Yurika!"（ ε ὕ ρηκα， The meaning is "found"
After further experiments, he arrived at the palace and placed the crown and an equal weight of pure gold in two pots filled with water. He compared the water overflowing from the two pots and found that the pot containing the crown overflowed more water than the other. This indicates that the volume of the crown is larger than that of pure gold of the same weight, with different densities, thus proving the inclusion of other metals in the crown.
The significance of this experiment far exceeds the discovery that the goldsmith deceived the king. Archimedes discovered the law of buoyancy (Archimedes principle) from it: the buoyancy obtained by an object in a liquid is equal to the weight of the liquid it discharges. (i.e. the well-known drainage method)

Lever principle
Principle of leverage: A system that satisfies the following three points is essentially a lever: fulcrum, force application point, and force application point. The principle of leverage, also known as the "lever balance condition": In order to balance a lever, the two torques acting on the lever (the product of the force and the lever arm) must be equal in magnitude. Namely: Power × Power arm=resistance × The resistance arm can be expressed as:
(F1 represents power, l1 represents power arm, F2 represents resistance, and l2 represents resistance arm)
King Hyveron encountered another tricky problem: the king built a ship for King Ptolemy of Egypt, but it was too large and heavy to be put into the sea, The king said to Archimedes, "You can even lift the Earth, so it shouldn't be a problem to put a ship into the sea. Archimedes asked craftsmen to install a set of intricately designed pulleys and levers on the front, back, left, and right sides of the ship. Archimedes asked more than 100 people to grab a rope in front of the ship, and he asked the king to pull it. The ship slowly slid into the sea. The king was extremely happy and announced in public: "From now on, I demand that everyone believe in Archimedes no matter what he says!"
Mechanical applications
Archimedes' research on machinery originated from his time studying in Alexandria. One day, while taking a walk by the drought stricken Nile River, he saw that farmers had to work very hard to water their fields. After thinking about it, he invented a tool that used the spiral action to rotate in a water pipe and suck water up. Later generations called it the Archimedes spiral water pump. Until modern times two thousand years later, people still used this instrument in Egypt. This tool became the ancestor of later spiral propellers.
Archimedes attached great importance to experimentation and designed and manufactured many instruments and machinery throughout his life, including lifting pulleys, irrigation machines, water pumps, and military stone throwers.
At that time, in Europe, simple machinery such as screws, pulleys, levers, gears, etc. were often used in engineering and daily life. Archimedes spent a lot of time researching and discovered the concepts of "lever principle" and "torque". For Archimedes, who often used tools to make machinery, it was easy to apply theory to practical life. Archimedes was likely the person with the most thorough understanding of the principles and applications of machinery in the world at that time.
There is a clear difference between Archimedes and the scientists of the Athenian period, which is that he not only valued the rigor and accuracy of science, but also demanded precise and logical proof for every problem; We also attach great importance to the practical application of scientific knowledge.

A master of mathematics
Archimedes also had brilliant achievements in mathematics, especially in geometry.
Archimedes' mathematical ideas contain calculus, and his Methodology is already very close to modern calculus. Here, there is an advanced study of "infinity" in mathematics, and throughout the entire article, it is about how to apply mathematical models in physics.
What he lacked was the concept of limits, but its essence extended to the mature field of infinitesimal analysis in the 17th century, foreshadowing the birth of calculus.
Archimedes effectively applied the convergence concept proposed by Euclid. He used the "approximation method" to calculate the area, volume, parabola, and elliptical area of the sphere, and later mathematicians developed modern "calculus" based on this "approximation method". Archimedes also used the circle cutting method to obtain a value of π between 3.14163 and 3.14286.
In addition, he calculated that the surface area of the ball is four times the maximum area of its inscribed circle, and derived that the volume of a sphere inscribed on a cylinder is two-thirds of its volume. This theorem is engraved on his tombstone. [7]
Archimedes studied the properties of spiral curves, and the current Archimedean spiral curve is named in honor of him. In addition, in his book "The Counter of Sand", he created a set of methods for counting large numbers, simplifying the way of counting.
Archimedes' geometric works were the pinnacle of Greek mathematics. He harmoniously combined Euclid's rigorous reasoning method with Plato's vivid and rich imagination, achieving a state of perfection and perfection, thus "making calculus, which was further cultivated by Kepler, Cavalieri, Fermat, Newton, Leibniz, and others, increasingly perfect".

Astronomical research
Archimedes developed a cross goniometer for astronomical measurements and created an instrument to measure the angle of the sun towards the Earth.
Archimedes also used hydraulic power to create a celestial globe with the sun, moon, stars, and five major planets on its surface. According to records, this celestial globe not only operates accurately, but can also predict when lunar or solar eclipses will occur.
Archimedes also believed that the Earth may be round. In his later years, Archimedes began to doubt the theory of the center of the Earth and speculate that it was possible for the Earth to rotate around the Sun. This speculation was not discussed until the Copernican era.